# Wake Analysis

5.1.1 Geometry of the Rotor Flow

5.1.1.1 Sweep Angle

The combination of rotation and forward speed makes the local inflow distribution across a helicopter rotor disc complex. One aspect is the angle of inflow at a given point of a rotor blade at a given azimuth angle; that is, the sweep angle. This section derives, in closed form, the equation defining the sweep angle contours.

Derivation of Contours

The rotor dimension is normalized to a unit radius. Also, the velocity components are normalized on the rotor tip speed. A general point of a rotor blade is at a normalized radius of (xRAD) and azimuth angle (C).

The velocity components defining the local sweep angle are shown in Figure 5.9.

From the figure, the sweep angle is defined by:

m cos C

tan w =

XrAD + m sin C

In order to determine the contour equations we use the following axes system (see Figure 5.10): the X axis (abscissa) lies in the incident airflow due to forward speed, that is over the tail, and they axis (ordinate) lies to starboard. With this axis system, the relationship between the normalized rotor radius (xRAD) and the azimuth angle (C) is as shown in Figure 5.11.

The transformation equations are:

XRAD = J X2 + У2

cos C = X/XRAD sin C = У/XRAD

Substituting (5.25) into (5.24) and clearing fractions gives:

y x

xrad + m———— m • cot w——

xRAD xRAD

hence:

x2+y2—m • cot w • x + my = 0 The equation of a general circle is given by:

where the centre of the circle is:

(—-f)

and the radius is:

Vg2 +f2—c

From (5.27)-(5.29), the circle has centre:

(m-w-—m)

and radius: which reduces to:

csc w

Geometrical Construction

This can be shown geometrically as in Figure 5.12.

The construction begins with drawing a line parallel to the flight direction (x) positioned at the distance of half the advance ratio; this is in fact passing through the centre of the reverse flow region. A second line is constructed from the origin (O, the centre of the rotor) at an angle to the

abscissa equal to the sweep angle contour required (w). This intersects the first construction line at point P. Using P as the centre and OP the radius, a circle is drawn. The part of this circle within the rotor disc (the unit circle) is the required contour. An example of these contours is shown in Figure 5.13 for an advance ratio of 0.25.

Examination of the equation governing the sweep angle contours enables the following conclusions to be made:

• The sweep angle contour is a circle.

• The zero angle contour is the advancing blade (C = 90°) and the retreating blade (C = 270°).

1

0.8 – 0.6 0.4 0.2 0

-0.2 -0.4 -0.6 -0.8 – -1

• The 90o contour is the periphery of the reverse flow region.

• All the contours pass within the reverse flow region. The sweep angle remains the same, but the flow direction is completely reversed.

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